Entire functions and potentials: Zeros of the Zeta function as redundant poles

✍ Scribed by N.N Khuri

• Publisher: Elsevier Science
• Year: 1990
• Tongue: English
• Weight: 657 KB
• Volume: 202
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(90)90342-l

✦ Synopsis

We prove the existence of a three dimensional potential, UR(r), whose S-wave scattering amplitude has the complex zeros of the Riemann Zeta function as "redundant poles." The potential is a C" function on O<r < co, and JR r lUR(r)I exp(ar)dr < co, for O<a<21,, where Ai is the ordinate of the first zero of the c-function. Detailed properties of U,(r) are given, and it is shown that its Fourier transform, 8,(t), is meromorphic in T, and the poles and residues are determined. The method is generalized to include a large class of even entire functions. The resulting potentials lead to scattering amplitudes with the zeros of the entire function as redundant poles. 0 1990 Academic PRESS, IIIC. * This paper is dedicated to Andre Martin on the occasion of his 60th birthday.